On the Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

Abstract

The theoretical prediction of the stochastic property of the quantum cosmological constant and the quantum stochastic nature of event horizons has crucial implications on the physics of space-time and black holes in particular. One of these consequences concerns a new mechanism, which is investigated here, for the stochastic regularization of singular black-hole solutions of classical general relativity. The problem is posed in the context of the theory of covariant quantum gravity (CQG-theory), namely the manifestly covariant, constraint-free and finite graviton-mass quantum Hamiltonian approach developed by Cremaschini and Tessarotto (2015–2022), which permits to cast the theory in a frame-independent setting. It is precisely the trajectory-dependence feature of the theory and the intrinsic stochastic property of quantum gravity which turn out to be crucial properties for reaching quantum regularization of classical singular solutions.

1. Introduction

This paper is a contribution to the current debate about quantum regularization of singular black hole (BH) classical solutions of the Einstein field equations and the investigation of their quantum stochastic characters. Despite recent interesting developments [1,2], the current prevailing view on the subject is that—purely from the quantum viewpoint—the determination of the possible quantum mechanism/interaction which may be responsible for the regularization of BH singularities is still missing. That the problem is a challenging and highly meaningful one emerges from two requirements:

• The first one is that all BH’s might/should actually manifest a central singularity. In fact, consistent with a quantum description of BH’s, instantaneous (i.e., superluminal) action-at-a-distance or mean-field forces should necessarily be omitted.
• The second requirement is that possible candidate quantum interactions should not violate the fundamental principles of quantum gravity (QG), i.e., in particular the principle of relativity on the speed of light in vacuum, and that of general relativity (frame independence), thus assuring that the theory remains frame—as well as background—independent.

The construction of regular BH’s in the framework of GR, i.e., based on classical physics models, is well known. As a possible example [3], they can be reached based on an anzatz for the 4-scalar space-time line element of the type (similar to the Schwarzschild solution)

$$d{s}^2=-f\left(r\right)d{t}^2{c}^2+\frac{d{r}^2}{f\left(r\right)}+r^{2}d{\Omega }^2,$$

(1)

where the so-called metric function f takes the form $f\left(r\right)=1-\frac{r_s\left(r\right)}{r}$ and $r_s\left(r\right)$ is an r-dependent “Schwarzschild-like” radius. To exclude a space-time singularity for $r\to 0$ it is sufficient to require that $r_s\left(r\right)$, $r_s^{\prime }\left(r\right)$ and $r_s^{″}\left(r\right)$ all vanish for $r\to 0$, while $r_s^{‴}\left(r\right)$ is finite in the same limit (which manifestly occurs if $r_s\left(r\right)/r^3$ remains finite in the same limit). It should be stressed also that such classical regular solutions cannot be ruled out ’a priori’, i.e., in the absence of a suitable QG theory.

Instead, regarding in particular the quantum regularization problem, we intend to propose a possible solution based on quantum gravity [4,5]. The theoretical setting is provided here by the so-called manifestly covariant approach to quantum gravity (CQG-theory [6,7]). This realizes a unitary representation of quantum theory for massive gravitons, which is ontologically similar (i.e., it is based on similar axioms) to quantum mechanics (QM) and is represented in a 4-tensor representation. However, in difference with QM, the same tensor representation is prescribed with respect to a suitable curved space-time, i.e., a 4-dimensional continuum background space-time whose choice, nevertheless remains arbitrary. In particular, in analogy to the Schrödinger equation, the corresponding quantum 4-scalar wave equation of CQG-theory, as well as the related 4-scalar quantum wave function, can similarly be parametrized in terms of statistical ensembles of configuration-space trajectories (the so-called GLP or generalized Lagrangian paths) and corresponding space-time trajectories of the form $r=r\left(s\right)$, which are identified with deterministic geodetics of the background space-time, parametrized with respect to the corresponding proper time s.

The goal of the paper is to attempt to reach possible new conclusions on the subject of quantum regularization based on the recent formulation of the so-called stochastic approach to CQG-theory, established on the basis of a Lagrangian approach to quantum gravity, which is fully manifestly covariant and background-independent. We refer the interested reader to the last reference for additional motivations and comparisons with current literature approaches to quantum gravity. As described in detail in the same reference, the newly added feature concerns the inclusion of stochastic (rather than deterministic) space-time trajectories, i.e., by means of the replacement of the form

$$r\left(s\right)\to r^{\left(q\right)}\left(s\right)\equiv r(s,\alpha ),$$

(2)

with $r\left(s\right)$ and $r^{\left(q\right)}\left(s\right)\equiv r(s,\alpha )$ representing, respectively, deterministic and stochastic space-time geodetics. These are both parametrized with respect to the proper time s, which is a 4-scalar, and the second one also in terms of the additional 4-scalar parameter $\alpha ,$ which is a constant and independent stochastic real parameter.

However, this feature should be introduced without affecting the crucial property of frame-independence characteristic of the original theory. In other words, it should still be true that the stochastic formulation of the theory should not depend on the choice of the GR-frames, to be prescribed with respect to the underlying background space-time. In order to warrant such a feature in QG—and keep also the frame-independent feature of CQG-theory [6,7]—the stochastic parametrization should not depend on the particular choice of the coordinates. This means that, purely on physical grounds, the only available choice requires introducing, for the deterministic proper time s the replacement with the stochastic proper time $s_1$ of the form

$$s\to s_1=s_1(s,\alpha ),$$

(3)

where by assumption both $s_1$ and $\alpha$ are 4-scalar real fields, and the second one represents also a suitably-defined stochastic variable. As a consequence, the stochastic trajectory $r^{\left(q\right)}\left(s\right)$ becomes necessarily the compound function

$$r^{\left(q\right)}\left(s\right)\equiv r\left(s_1(s,\alpha )\right).$$

(4)

Regarding the physical meaning of Equation (4) we notice that, in practice, it means that, while the trajectory itself remains unchanged, both the positions at the initial time (defined at some initial $s=s_0$) $r^{\left(q\right)}\left(s_0\right)=r\left(s_1(s_0,\alpha )\right)$ and at a generic proper time s become stochastic and are displaced with respect to the corresponding deterministic positions $r\left(s_o\right)$ and $r\left(s\right)$. A brief reminder is necessary concerning the choice of CQG-theory to address the problem of resolving classical singularities of the background metric tensor. We notice in this regard that the rationale of CQG-theory finds its roots already in the establishment of a suitable Hamiltonian representation of GR, i.e., for the Einstein field equation (EFE). However such a representation is nonunique. Indeed, it is well known that in GR there are (at least) two different possible Hamiltonian representations available that may be invoked for achieving a quantum theory of the gravitational field. The possible alternatives are provided respectively either by a constrained and nonmanifestly covariant or by an unconstrained and manifestly covariant classical abstract Hamiltonian systems. Such choices lead to qualitatively different quantum gravity (QG) theories. The first case corresponds in particular to the framework provided by the so-called quantum geometrodynamics (QGD), to be distinguished in its two main variants, namely the Wheeler–DeWitt (WDW) equations [8] of QGD and the so-called loop quantum gravity (LQG) (for a review, see [9]).

However, it is only the unconstrained and manifestly covariant Hamiltonian representation of GR which permits to achieve a so-called manifestly covariant representation of QG, namely which is cast in 4-tensor form with respect to a 4-dimensional background space-time. The latter is taken of the form $\left\{{\mathbf{Q}}^4,\widehat{g}\left(r\right)\right\}$, with ${\mathbf{Q}}^4\subseteq {\mathbb{R}}^4$ being a 4-dimensional time-oriented (background) Lorentzian space-time and $\widehat{g}\left(r\right)=\left\{{\widehat{g}}_{\mu \rho }\left(r\right)\right\}=\left\{{\widehat{g}}^{\mu \nu }\left(r\right)\right\}$ representing an appropriate quantum-modified background metric tensor. Its characteristic feature—which turns out to be crucial for the purposes of the present paper—is the trajectory dependence of the theory, namely the fact that all quantum fields, including the quantum wave function, are parametrized with respect to the (quantum) trajectories of quantum gravitons. These are the primary reasons for adopting CQG-theory rather than QGD (in either of its variants). In particular, one can show that in the context of CQG-theory, the quantum-modified background metric tensor $\widehat{g}\left(r\right)$ is determined self-consistently and is identified with a particular solution of the so-called quantum-modified EFE. Thanks to manifest covariance also such an equation—just as the original EFE holding in GR—is represented in explicit 4-tensor form and hence is frame-independent. For additional useful features of CQG-theory we refer to the original formulation given in refs. [6,7], the discussion of the Heisenberg uncertainty principle [10,11], and the recent contributions on quantum logic [12,13] where it was shown that CQG-theory admits a quantum logic analogous to that of quantum mechanics.

Nevertheless, all these features do not come without price, in the sense that they imply also an obvious underlying condition for the validity of CQG-theory. In fact the same setting holds everywhere in $\left\{{\mathbf{Q}}^4,\widehat{g}\left(r\right)\right\}$ only provided $\widehat{g}\left(r\right)$ is defined everywhere, i.e., there are no singular points in $\left\{{\mathbf{Q}}^4,\widehat{g}\left(r\right)\right\}$. The reason emerges perspicuously by direct inspection of the quantum wave function $\psi \left(s\right)\equiv$$\psi (g,\widehat{g},r\equiv r\left(s\right),s)$ in the context of CQG-theory. In particular, let us introduce the Madelung representation

$$\psi \left(s\right)=\sqrt{\rho \left(s\right)}exp\left\{\frac{i}{\hslash }S\left(s\right)\right\},$$

(5)

with $\rho \left(s\right)$ and $S\left(s\right)$ denoting, respectively, the quantum probability density function (PDF) and the quantum phase function, so that by construction $\psi \left(s\right){\psi }^{*}\left(s\right)=\rho \left(s\right)$. Then, one finds out that the quantum PDF $\rho \left(s\right)$ written in the so-called GLP-representation [14] takes the form

$$\rho \left(s\right)=Kexp\left\{-\frac{\left(\Delta g(s\right)-\widehat{g}{\left(r\right))}^2}{r_{th}^2}\right\},$$

(6)

where K and $r_{th}^2$ are dimensionless 4-scalars, $\widehat{g}\left(r\right)\equiv \widehat{g}\left(r\left(s\right)\right)$ is the extremal background tensor and the exponential factor $\left(\Delta g(s\right)-\widehat{g}{(r,s))}^2$ denotes $\left(\Delta g(s\right)-\widehat{g}{\left(r\right))}^2\equiv \left(\Delta g(s\right)-\widehat{g}\left(r\right){{)}_{\mu \nu }\left(\Delta g(t,s\right)-\widehat{g}\left(r\right))}^{\mu \nu }$, with $\Delta g\left(s\right)$ being a stochastic symmetric tensor. We stress that, according to the notation introduced earlier in Ref. [6], in the present context, the term “extremal” refers to the condition whereby the variation of a given tensor quantity vanishes identically. Therefore, the same term must not be confused hereon with the meaning that it acquires in black hole theory in reference to special realizations of charged and rotating black holes. Thus, let us assume for definiteness that ${\widehat{g}}_{\mu \nu }\left(r\right)$ diverges (i.e., is not defined) at some point $r=r^{*}$ of the space-time ${\mathbf{Q}}^4$. Then, if $\Delta g\left(s\right)$ identifies an arbitrary stochastic tensor with nonvanishing components at the same point $r=r^{*}$, it generally follows that ${\widehat{g}}_{\mu \nu }\left(r\right)\Delta g^{\mu \nu }\left(s\right)$ may not be defined at the same point $r=r^{*}$. This implies therefore that the same exponential factor, as well as both the quantum probability density $\rho \left(s\right)$ and the quantum wave function $\psi \left(s\right)$ are generally not defined at $r=r^{*}$ (in view of the Madelung representation indicated above (5)). This means that in order to warrant the global validity of CQG-theory, the background metric tensor must be necessarily regular in the whole space-time domain, requiring in turn that if black holes (BH) exist, they must be also regular within the same quantum description provided by CQG-theory. This implication follows from the fact that the quantum PDF carries itself the background metric tensor ${\widehat{g}}_{\mu \nu }\left(r\right)$, which realizes consistently the property of emergent gravity. Accordingly, the same tensor ${\widehat{g}}_{\mu \nu }\left(r\right)$ can be interpreted as arising from an appropriate statistical average of the stochastic tensor $\Delta g$ performed over the quantum PDF on integration domain of variational fields $U_g$, namely of the type

$${\widehat{g}}_{\mu \nu }\left(r\right)=\underset{U_g}{\int }d(\Delta g)\Delta g_{\mu \nu }\rho \left(s\right).$$

(7)

It follows that, for the validity of the theory and its formalism, there must be a mutual consistency condition of existence between $\rho \left(s\right)$ and ${\widehat{g}}_{\mu \nu }\left(r\right)$, so that for the global validity of the former function, the latter tensor must itself prevent the occurrence of space-time singularities.

The question to be investigated below is therefore how such a property can actually be warranted in the context of CQG-theory.

2. Regularization and Stochastic Effects in CQG-Theory

The problem of the regularization of the background metric tensor $\widehat{g}\left(r\right)$ has been already discussed in previous papers developed in the same conceptual framework represented by CQG-theory. More precisely, in Ref. [1] the issue was tackled implementing a conformal solution, namely by replacing the same metric tensor $\widehat{g}\left(r\right)$ with a conformal representation in terms of a 4-tensor of the form

$${\widehat{g}}_{\mu \nu }^{\left(d\right)}(r,s)=N\left(s\right){\widehat{g}}_{\mu \nu }\left(r\right).$$

(8)

Then, it was shown that the 4-scalar scale-function $N\left(s\right)$ can be self-consistently determined in such a way that the conformal (i.e., scale-transformed) tensor ${\widehat{g}}_{\mu \nu }^{\left(d\right)}\left(r\right)$ actually may exhibit a regular behavior in the limit $r\to r^{*}$. Nevertheless, it is obvious that since the countervariant components take the form

$${\widehat{g}}^{\left(d\right)\mu \nu }(r,s)=\frac{1}{N\left(s\right)}{\widehat{g}}^{\mu \nu }\left(r\right),$$

(9)

they continue to be singular at $r=r^{*}$. In other words, in a sense, the singularity is actually not ruled out.

On the other hand, in Ref. [2] it has been shown that the event horizon (EH) of a Schwarzschild–deSitter BH, and in principle of arbitrary BH’s as well, may be characterized by a fundamental stochasticity property. The crucial feature that emerges is that the effective radius of the inner event horizon $r_S^{\left(E\right)}$, characterizing in that case the Schwarzschild–deSitter solution, can actually exhibit a stochastic property too. In other words, it takes the form of a linear stochastic shift of the form

$$r_S^{\left(E\right)}=r_S\pm r_o^{\left(\mathsf{\Lambda }\right)},$$

(10)

with

$$r_S=2GM/c^2$$

(11)

being the Schwarzschild radius and $r_o^{\left(\mathsf{\Lambda }\right)}$$>0$ representing a stochastic contribution associated with the cosmological constant (CC) predicted by CQG-theory [2].

We stress that the assumption (10) is perturbative in character. However, as shown in Ref. [2], the stochastic nature of the CC is an intrinsic feature of CQG-theory built-in into the quantum formalism. In this regard a number of unique aspects of the same theory must be emphasized:

• The first basic feature is the nonperturbative character of CQG-theory [6,7], in which, additionally, the background metric tensor is determined self-consistently [14].
• The second feature is that the physically observed CC can actually be explained in terms of a purely quantum phenomenon arising in CQG-theory. As shown in the same reference, the physical explanation that emerges is that the CC arises solely due to the so-called Bohm–vacuum interaction that arises among gravitons in vacuum.
• The third feature of CQG-theory is that the initial quantum PDF is permitted to be intrinsically stochastic in character. As a consequence, the quantum CC itself (see Equation (36) below) becomes stochastic in its own right.
• The fourth feature is that the stochasticity of the CC implies also a stochastic shift of the event horizon. This means that quantum gravitons can effectively cross EH’s because they are no more impenetrable barriers.
• Finally, as shown in Ref. [2], the same stochastic shift can be approximated in terms of the linear asymptotic representation provided by Equation (10). The linearization (10), however, is not fruit of some kind of arbitrary “ad hoc” approximation, but, on the contrary, a feature following from the fact that the radius of BHs is typically much smaller than that of the deSitter EH.

In other words, at the physical level it follows the fundamental physical consequence that the classical Schwarzschild radius $r_S$ is effectively subject to a stochastic quantum shift of magnitude $r_o$, whose signature remains in principle to be determined, so that the classical impenetrable barrier at $r_S$ is replaced by a quantum belt having an approximate width $2r_o$ where the quantum stochastic properties of the EH take place. The same behavior has been equivalently predicted to occur also at the cosmological horizon of the deSitter solution [2]. This suggests that a similar stochastic quantum shift of the center of symmetry of the BH might occur. It means that the spherical coordinate of the center $r^{*}=0$, to be identified with the origin point of the Schwarzschild sphere, should undergo a corresponding infinitesimal stochastic radial shift of the type

$$r^{*}\to r^{*}+r_o,$$

(12)

with $r_o$ representing an outward stochastic shift of the BH’s center of symmetry. Notice here that $r_o$ may not necessarily coincide with the analogous one indicated above, $r_o^{\left(\mathsf{\Lambda }\right)}$. The conjecture, however, does not suggest by itself what might be a possible physical mechanism for the quantum regularization of Schwarzschild-type singularities. We intend now proceed to investigate whether the explicit construction of such a stochastic transformation is actually possible and physically admissible.

3. The Heisenberg Indeterminacy Principle

However, a further nontrivial feature seems worth to be mentioned. This refers to the Heisenberg indeterminacy principle which holds in CQG-theory [10] also in its “generalized form” [11], which involves the adoption of a nonlocal scalar product (notice, however, that this feature does not make the theory itself nonlocal since all quantum variables remain intrinsically local in space-time, as required for covariance). This makes possible the treatment of a “generalized Heisenberg inequality”, i.e., an inequality, which involves the extended conjugate canonical variables $\left(s,p_s^{\left(q\right)}\right)$, namely the proper-time s and its extended conjugate canonical momentum, which is given by the quantum operator

$$p_s^{\left(q\right)}\equiv -i\hslash \frac{\partial }{\partial s}.$$

(13)

We stress that here s is the proper length defined on an arbitrary geodetics of the background metric tensor $\widehat{g}\left(r\right)\equiv \left\{g_{\rho \nu }\left(r\right)\right\}\equiv \left\{g^{\rho \nu }\left(r\right)\right\}$. As a consequence it is a 4-scalar. According to the same reference, the said inequality is provided by

$${\sigma }_s{\sigma }_{p_s^{\left(q\right)}}\ge \frac{\hslash }{2},$$

(14)

which realizes the proper-time-extended canonical momentum (extended) Heisenberg inequality. Therefore, the simultaneous quantum measurement of proper time s and its conjugate quantum extended momentum $p_s^{\left(q\right)}$ during the proper time interval $\left(s_o,s_1=s_0+\Delta s\right)$ involves the evaluation of the expectation values $\left(\tilde{s},{\tilde{p}}_s^{\left(q\right)}\right)$, together with the related standard deviations ${\sigma }_s$ and ${\sigma }_{p_s^{\left(q\right)}}$. In particular, ${\tilde{p}}_s^{\left(q\right)}$ is related to the expectation value of the quantum Hamiltonian and thus defines an invariant energy. The corresponding quantum fluctuations (i.e., the squares of the standard deviations ${\sigma }_s$ and ${\sigma }_{p_s^{\left(q\right)}}$) are therefore

$$\begin{array}{ccc}\hfill {\sigma }_s^2& \equiv & 〈〈\psi \left|{(s-\tilde{s})}^2\psi \right.〉〉,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\sigma }_{p_s^{\left(q\right)}}^2& \equiv & 〈〈\psi \left|{\left(p_s^{\left(q\right)}-{\tilde{p}}_s^{\left(q\right)}\right)}^2\psi \right.〉〉,\hfill \end{array}$$

(16)

with $\psi \left(s\right)\equiv$$\psi (g,\widehat{g},r\equiv r\left(s\right),s)$ being the quantum wave-function. In the previous equations, the symbol “$〈〈•\left|•\right.〉〉$” denotes the nonlocal scalar product [11] whose definition is recalled in Appendix A (see Equation (A1)), while $〈〈{\psi }_a\left|{\psi }_b\right.〉〉$ is its realization obtained letting ${\psi }_a\equiv \psi$ and identifying ${\psi }_b$, respectively, either with ${\psi }_b\equiv {(s-\tilde{s})}^2\psi$ or ${\psi }_b$$\equiv {\left(p_s^{\left(q\right)}-{\tilde{p}}_s^{\left(q\right)}\right)}^2\psi$. Then, formally setting $s_0=$ 0 in the time averages, one finds out that the expectation value and standard deviation of s are, respectively

$$\begin{array}{ccc}\hfill \tilde{s}& =& \frac{1}{\underset{s_0}{\overset{s_1}{\int }}ds}\underset{s_0}{\overset{s_1}{\int }}dss=\frac{1}{\underset{0}{\overset{\Delta s}{\int }}ds}\underset{0}{\overset{\Delta s}{\int }}dss=\frac{\Delta s}{2},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\sigma }_s& \equiv & \sqrt{〈〈\psi \left|{(s-\tilde{s})}^2\psi \right.〉〉}=\frac{\Delta s}{2\sqrt{3}}.\hfill \end{array}$$

(18)

The last equation relates the standard deviation ${\sigma }_s$ of the proper time s to the amplitude of the proper time interval $\Delta s$ (an invariant length), during which the quantum measurement is being performed. This implies that, written in terms of $\Delta s$, the previous inequality yields:

$$\Delta s{\sigma }_{p_s^{\left(q\right)}}\ge \hslash \sqrt{3}.$$

(19)

The inequality (19) implies a number of interesting consequences:

• First, $\Delta s$ can be interpreted as a minimal length associated with the quantum measurement of the canonical momentum $p_s^{\left(q\right)}$, with the immediate consequence that CQG-theory can be viewed as a minimal length theory.
• Its existence is not at variance with the validity of CQG-theory and in particular its manifest covariance.
• The same minimal length $\Delta s$ is an invariant length, i.e., a 4-scalar with respect to the background space-time of CQG-theory. The invariance character of $\Delta s$ follows from the fact that the latter is a measure of proper-time interval, and therefore of arc-length, while the same proper-time is a 4-scalar of the background space-time.
• Since the expectation value ${\tilde{p}}_s^{\left(q\right)}$ is an invariant energy, one expects the standard deviation to be related to quantum fluctuations of the same invariant energy. The precise minimum value of $\Delta s$ depends, however, on the standard deviation associated with the same quantum canonical momentum.
• Finally, provided ${\sigma }_{p_s^{\left(q\right)}}$ is nonvanishing, then $\Delta s$ is strictly positive and therefore the limit $\Delta s\to 0$ becomes physically meaningless.

These features, besides providing a possible alternative view regarding the concept of minimal length usually adopted in the framework of phenomenological generalized uncertainty principle (GUP) theories [15,16,17,18,19], may have also a possible connection with the BH quantum regularization problem considered here. The conjecture, in fact, is that the invariant minimal length $\Delta s$ might possibly suggest a way out for the regularization problem of classical singular BH solutions. In fact, if one introduces, for definiteness, a local spherical coordinate system at rest with the BH (which is assumed stationary), one might assume without loss of generality the said center of symmetry, where the central singularity occurs, to have the radial coordinate $r(s_o=0)\equiv 0$. The conjecture is that the invariant minimal length $\Delta s$ (as determined by the inequality (19)) might be associated with a shift in the radial coordinate

$$r(s_o=0)\equiv 0\to r_o\equiv r(s_1=\Delta s)>0,$$

(20)

being $r_o\equiv r(s_1=\Delta s)$ a nonvanishing radial coordinate shift, representing an arbitrary point sufficiently close to $r(s_o=0)\equiv 0$ in the BH interior, where, however, the background metric tensor is regular. The radial shift (20) might be interpreted as a displacement arising in the quantum trajectories near the center of symmetry of the BH as an effect of stochastic corrections. This suggests therefore that a possible stochastic modification of CQG-theory might be appropriate.

4. Stochastic Quantization

It is appropriate at this point to emphasize the main reasons supporting the need of implementing a stochastic quantization. Indeed, it is well-established in quantum mechanics (QM) that, when adopting a trajectory-based representation, all quantum particle trajectories should be nondeterministic (i.e., stochastic) in character. This concept goes against the customary, and for this reason physically incorrect, Bohmian interpretation of QM. Although physically intuitive the possible motivations need an explanation. Occurrence of stochastic space-time quantum trajectories can arise in principle for two main reasons:

• The first one concerns the intrinsic stochastic character of space-time quantum trajectories. Their identification with deterministic trajectories, as performed in the original formulation of Bohmian quantum mechanics is only an approximation. In fact, in view of the discussion given in the previous section this may give rise to a violation of the Heisenberg indeterminacy principle holding in CQG-theory. In the case of QM, the proof of the stochastic character of quantum trajectories was reached rigorously in Ref. [20]. In analogy to QM, it appears therefore reasonable to expect that trajectories of graviton particles that belong to the background space-time should similarly possibly acquire a stochastic character and therefore should depart from deterministic geodetic curves [6,7].
• The second reason is the spatially extended feature of quantum particles. In fact, all quantum particles (including graviton particles), in a strict sense, should be treated as spatially extended and not just as point-like particles. There are at least two possible physical motivations: (a) the proper treatment of particle self-interactions: in the case of point particles, they cannot be properly defined (see for example Ref. [21] where the case of electromagnetic self-interactions was considered) or (b) in the present context, the issue of possible quantum regularization of space-time singularities that occur thanks to the stochastic property of particle trajectories.

However, independent of the stochastic trajectories, there is also another type of stochasticity, which must be taken into account since it brings an important physical effect:

• This is related to the possible occurrence of explicit stochastic gauge contributions in the quantum Hamiltonian operator. These contributions—as discussed in Section 4—become crucial for the explicit construction in terms of quantum averages of the quantum-modified EFE. As a consequence, this type of stochastic gauge contributions should be actually regarded as mandatory for the prescription of the background metric field tensor $\widehat{g}\left(r\right)$.

The resulting new scheme of canonical quantization for GR, to be denoted as stochastic quantization is based on the introduction of two transformations, referred to as stochastic quantizations transformations. More precisely:

(A)

The first one is related to the inclusion of stochastic quantum trajectories. Since CQG-theory represents a trajectory-dependent theory, this effectively requires replacing the classical (i.e., deterministic) space-time trajectories $\left\{r\left(s\right)\right\}$ with stochastic quantum trajectories $\left\{r^{\left(q\right)}\left(s\right)\right\}$. Concerning the functional dependence, in this notation the parameter s identifies the 4-scalar proper-time determined by the differential relation $d{s}^2={\widehat{g}}_{\mu \nu }d{r}^{\mu }d{r}^{\nu }$ holding along a subset of classical geodesic curves. Thus, stochastic quantization must necessarily involve the additional synchronous transformation

$$\left\{r\left(s\right)\right\}\to \left\{r^{\left(q\right)}\left(s\right)\right\},$$

(21)

which leaves unaffected the proper time s, where

$$r^{\left(q\right)}\left(s\right)\equiv r(s,\alpha )$$

(22)

is a stochastic (quantum) trajectory which, in accordance with Equation (4), reads:

$$r(s,\alpha )\equiv r\left(s_1(s,\alpha )\right).$$

(23)

Notice that $\delta r=r(s,\alpha )-r\left(s\right)$ represents therefore the stochastic shift to be placed on the geodetic curve $\left\{r\left(s\right)\right\}$ associated with the classical background metric tensor $\widehat{g}\left(r\right)$ [6]. By assumption the quantum trajectories $\left\{r(s,\alpha )\right\}$ are subject to a deterministic constraint of the form

$$\underset{\alpha \to 0^{\pm }}{lim}r(s,\alpha )=\underset{\alpha \to 0^{\pm }}{lim}r\left(s_1(s,\alpha )\right)=r\left(s\right).$$

(24)

Here, $\alpha$ denotes a dimensionless, so-called “hidden” variable, i.e., a stochastic independent 4-scalar parameter. Instead, $r(s,\alpha )$ is a space-time stochastic curve to be assumed such that the displacement $r(s,\alpha )-r\left(s\right)$ is suitably small with respect to the characteristic scale length of the geodesics $r\left(s\right)$. Its definition is intrinsically nonunique. However, for definiteness, one can assume in particular that $\alpha$ belongs to the finite set

$$I_{\alpha }=\left[-a,a\right]-\left\{-\epsilon ,\epsilon \right\},$$

(25)

where either $a=1,\infty ,$ the value $\alpha =0$ is assumed forbidden, while $\alpha$ is assumed to be endowed with a stochastic PDF, whose form remains largely arbitrary. Thus, possible examples are provided by: (a) a binomial PDF with $\alpha$ taking only the values $\pm 1$; (b) a Gaussian PDF of the form

$$g_{\alpha }\left(\epsilon \right)=Nexp\left\{-{\alpha }^2/{\epsilon }^2\right\}.$$

(26)

where $\epsilon >0$ is a dimensionless parameter such that $\epsilon \ll 1,$ and N is a normalization constant defined so that

$${〈1〉}_{\alpha }\equiv {\int }_{I_{\alpha }}d\alpha g_{\epsilon }\left(\alpha \right)=1.$$

(27)

Instead, ${〈•〉}_{\alpha }$ denotes the stochastic $\alpha -$ average

$${〈•〉}_{\alpha }\equiv {\int }_{I_{\alpha }}d\alpha •g_{\epsilon }\left(\alpha \right).$$

(28)

Thus, $\alpha$ is assumed to be endowed with a vanishing stochastic expectation value ${〈\alpha 〉}_{\alpha }=0$ and standard deviation ${\sigma }_{\alpha }=\sqrt{{〈{\alpha }^2-{〈\alpha 〉}_{\alpha }^2〉}_{\alpha }}\sim O\left({\epsilon }^2\right)$. It follows that, without loss of generality, $\alpha$ can always be assumed to coincide with the stochastic parameter introduced in Ref. [2], which defines the quantum cosmological constant ${\mathsf{\Lambda }}_{CQG}$ and determines its quantum-stochastic property.

(B)

The second quantization transformation arises because of new added stochastic contributions to the quantum Hamiltonian operator, which are required to vanish in the limit of classical Hamiltonian function. This is represented by a transformation of the type

$$H_R\left(g,\frac{\partial S}{\partial g},\widehat{g}\left(s\right),r\left(s\right)s\right)\to H_R^{\left(q\right)}(g,-i\hslash \frac{\partial }{\partial g},\widehat{g}\left(r(s,\alpha )\right),r(s,\alpha ),s)=H_R\left(g,-i\hslash \frac{\partial }{\partial g},\widehat{g}\left(r(s,\alpha )\right),r(s,\alpha ),s\right)+\Delta V,$$

(29)

where, for simplicity of notation, hereafter $H_R$, $H_R^{\left(q\right)}$ and $\Delta V$ represent, respectively, the classical Hamiltonian function of GR (we refer here to Equation (A3) in Appendix B for the relevant notations), the quantum Hamiltonian and a stochastic gauge contribution.

4.1. Stochastic Canonical Map and Stochastic CQG-Quantum-Wave Equation

The adoption of Equations (21) and (29) involves a reformulation of the canonical quantization scheme developed originally for CQG-theory. In detail, based on the Hamilton–Jacobi $g-$ quantization scheme first developed in Ref. [10], this is achieved by the prescription of the mapping which determines the 4-scalar quantum wave-function $\psi \left(s\right)$ in terms of the corresponding 4-scalar classical Hamilton principal function $S\left(s\right)\equiv S(g,\widehat{g},r\left(s\right),s)$:

$$S\left(s\right)\to \psi \left(s\right).$$

(30)

Notice that the transformation (21) applies to all fields. In particular, the background metric tensor $\widehat{g}\left(r\left(s\right)\right)$ and the background Ricci tensor, namely defined in terms of the same $\widehat{g}(r\left(s\right))$ as ${\widehat{R}}_{\mu \nu }=R_{\mu \nu }(\widehat{g}\left(r\left(s\right)\right)$, transform as

$$\begin{array}{ccc}\hfill \widehat{g}\left(s\right)& \equiv & \widehat{g}\left(r\left(s\right)\right)\to {\widehat{g}}^{\left(q\right)}\left(s\right)\equiv \widehat{g}\left(r(s,\alpha )\right),\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\widehat{R}}_{\mu \nu }& =& R_{\mu \nu }\left(\widehat{g}\left(r\left(s\right)\right)\right)\to {\widehat{R}}_{\mu \nu }^{\left(q\right)}\equiv R_{\mu \nu }\left(\widehat{g}\left(r(s,\alpha )\right)\right),\hfill \end{array}$$

(32)

while the canonical fields $x=\left\{g_{\mu \nu },{\pi }_{\mu \nu }\equiv \frac{\partial S(g,\widehat{g},r\left(s\right),s)}{\partial g^{\mu \nu }}\right\}$ transform according to the stochastic canonical quantization map (where for simplicity we use the same symbols for the quantum variables)

$$\left\{\begin{array}{c}g\left(s\right)\equiv g\left(r\right(s),s)\\ \pi \left(s\right)\equiv \pi (r\left(s\right),s)=\frac{\partial S(g,\widehat{g},r\left(s\right),s)}{\partial g}\\ H_R\left(g,\frac{\partial S}{\partial g},\widehat{g}\left(s\right),r\left(s\right)s\right)\end{array}\to \right.\left\{\begin{array}{c}{g}_{\mu \nu }\left(s\right)=g_{\mu \nu }(r(s,\alpha ),s)\\ {\pi }_{\mu \nu }\left(s\right)\equiv -i\hslash \frac{\partial }{\partial g^{\mu \nu }(r(s,\alpha ),s)}\\ H_R^{\left(q\right)}(g,-i\hslash \frac{\partial }{\partial g},\widehat{g}(r(s,\alpha ),r(s,\alpha ),s)\end{array}\right..$$

(33)

On the left-hand side of Equation (30), $S\left(s\right)\equiv S(g,\widehat{g},r\left(s\right),s)$ and the fields $(g,\widehat{g})=(g,\widehat{g}\left(r\left(s\right)\right))$ are evaluated at the Lagrangian coordinate g and along the same (generic) classical geodesics $\left\{r\left(s\right)\right\}$. Instead, on the rhs, $\psi \left(s\right)$ is assumed as a complex function of the type

$$\psi \left(s\right)\equiv \psi (g,\widehat{g}(r(s,\alpha ),r(s,\alpha ),s).$$

(34)

Notice that now—in difference with Ref. [10]—for an arbitrary Lagrangian coordinate g, the function $\psi \left(s\right)$ is evaluated at the proper time s and along an (arbitrary) stochastic space-time trajectory $r^{\left(q\right)}\left(s\right)$. Furthermore, while in the classical Hamiltonian $H_R$ the canonical momenta $\pi$ are represented in terms of the Hamilton principal function by letting ${\pi }_{\mu \nu }=\frac{\partial S(g,\widehat{g},r\left(s\right),s)}{\partial g^{\mu \nu }}$, in the corresponding quantum Hamiltonian operator $H_R^{\left(q\right)}\left(g,\pi ,\widehat{g}(r(s,\alpha ),r(s,\alpha ),s\right)$ they are prescribed according to Equation (33) and again the same operator is evaluated with respect to the stochastic space-time trajectory $r^{\left(q\right)}\left(s\right)\equiv r(s,\alpha )$. Finally, the Hamilton–Jacobi quantization, which maps in each other the Hamilton–Jacobi equation and the corresponding quantum-wave equation, now becomes

$$\begin{array}{|c|}\hline \frac{dS}{ds}+H_R=0\\ \hline\end{array}\to \begin{array}{|c|}\hline \left[-i\hslash \frac{d}{ds}+H_R^{\left(q\right)}\right]\psi \left(s\right)=0,\\ \hline\end{array}$$

(35)

where the equation on the rhs is now referred to as stochastic CQG-quantum-wave equation. The same equation advances in time along the space-time stochastic trajectories (21) the stochastic quantum wave function $\psi \left(s\right)$, which is assumed of the type (34). Finally, we recall that in both of the above equations the operator $\frac{d}{ds}$ denotes the covariant $s-$ derivative which acts both on the explicit and implicit dependences contained through $r\left(s\right)$ and $r(s,\alpha )$, respectively, and is performed, instead, keeping constant the Lagrangian coordinate g (see notations reported in Appendix C).

4.2. Stochastic-Averaged Quantum-Modified Einstein Field Equations and Stochastic Regularization

At this point, following the guidelines of Ref. [2], the stochastic quantum-modified Einstein field equations can be formally recovered. Thus, in place of Equation (29) of Ref. [2], one can prove that in the stationary case (with respect to proper-time) and after taking the stochastic average ${〈•〉}_{\alpha }$, the CQG-wave equation now delivers:

$${〈R_{\mu \nu }\left(\widehat{g}\left(r(s,\alpha )\right)\right)〉}_{\alpha }-\frac{1}{2}{〈R\left(\widehat{g}\left(r(s,\alpha )\right)\right){\widehat{g}}_{\mu \nu }(r(s,\alpha ),s)〉}_{\alpha }={〈T_{\mu \nu }\left(\widehat{g}\left(r(s,\alpha )\right)\right)〉}_{\alpha }-{\mathsf{\Lambda }}_{CQG}{〈(1-\alpha )\widehat{g}\left(r(s,\alpha )\right)〉}_{\alpha },$$

(36)

which determines the so-called stochastic quantum-modified EFE. The remarkable aspect of such and equation is that all relevant tensor fields appear expressed in terms of stochastic averages and not in terms of their stochastic values. This feature may hopefully “cure” the singular behavior which typically affects the Einstein tensor field equation of classical GR.

On the other hand, the same equation, under smoothness conditions, i.e., far from the singularities, can be shown to be approximated asymptotically by the customary quantum-modified EFE earlier determined in the framework of CQG-theory (see [2]). The result follows by invoking for $r(s,\alpha )$ a Taylor expansion with respect to $\alpha$ which delivers an expansion of the form $r(s,\alpha )=r\left(s\right)+\alpha r_1\left(s\right)+{\alpha }^2{r}_2\left(s\right)$. Then, if all involved tensor fields are smooth functions of $r(s,\alpha )$, which do not locally diverge, one obtains ignoring corrections of $O\left({\epsilon }^2\right)$:

$$\begin{array}{ccc}\hfill {〈\widehat{g}\left(r(s,\alpha )\right)〉}_{\alpha }& \simeq & \widehat{g}\left(r\left(s\right)\right)\left[1+O\left({\epsilon }^2\right)\right],\hfill \\ \hfill {〈R_{\mu \nu }\left(\widehat{g}\left(r(s,\alpha )\right)\right)〉}_{\alpha }& \simeq & R_{\mu \nu }(\widehat{g}\left(r\left(s\right)\right)\left[1+O\left({\epsilon }^2\right)\right],\hfill \\ \hfill {〈R\left(\widehat{g}\left(r(s,\alpha )\right)\right){\widehat{g}}_{\mu \nu }(r(s,\alpha ),s)〉}_{\alpha }& \simeq & R(\widehat{g}\left(r\left(s\right)\right){\widehat{g}}_{\mu \nu }(r\left(s\right),s)\left[1+O\left({\epsilon }^2\right)\right],\hfill \\ \hfill {〈T_{\mu \nu }\left(\widehat{g}\left(r(s,\alpha )\right)\right)〉}_{\alpha }& \simeq & T_{\mu \nu }\left(\widehat{g}\left(r\left(s\right)\right)\right)\left[1+O\left({\epsilon }^2\right)\right].\hfill \end{array}$$

(37)

The fact, however, that the two equations do not coincide exactly is not unexpected. Indeed, Equation (36) retains the average effect of stochastic perturbations and therefore should be regarded as an exact one.

Now, let us assume for definiteness that one of the tensor components of $\widehat{g}\left(r(s,\alpha )\right)$, at $r(s,\alpha =0)=0$ contains a singularity. Thus, for definiteness, let us assume without loss of generality that, for example, the same singularity is of the type $\widehat{g}\left(r(s,\alpha )\right)\sim 1/r$ which diverges in the limit ${lim}_{r\to 0}$ (where here $r\equiv r(s,\alpha )$ identifies a radial coordinate in a frame locally at rest with respect to the singularity). Then, it follows that its stochastic average may actually happen to be regular, i.e., such that

$${〈\widehat{g}\left(r(s,\alpha )\right)〉}_{\alpha }={〈\widehat{g}(r\left(s\right)+\alpha r_1\left(s\right)+{\alpha }^2{r}_2\left(s\right))〉}_{\alpha }\sim {〈\frac{1}{r\left(s\right)+\alpha r_1\left(s\right)+{\alpha }^2{r}_2\left(s\right)}〉}_{\alpha }<\infty .$$

(38)

Similar considerations apply in principle for arbitrary power-law singularities of the form $\widehat{g}\left(r(s,\alpha )\right)\sim 1/r^{\beta }$ for $\beta >1$. As a consequence, the central singularity of an arbitrary BH singular solution may actually disappear once the stochastic averaging operator ${〈•〉}_{\alpha }$ is acted upon.

5. Regularization of Schwarzschild-Type Solutions

Let us now investigate in greater detail the behavior of the solutions of the stochastic quantum-modified EFE provided by Equation (36). Let us consider for definiteness the vacuum case, ignoring infinitesimal contributions and in particular ignoring the effect of the quantum cosmological constant ${\mathsf{\Lambda }}_{CQG}.$ We assume in particular that in Equation (36) the relative magnitude of the rhs terms is such that

$${\mathsf{\Lambda }}_{CQG}\left(s\right)(1-\alpha ){\widehat{g}}_{\mu \nu }\left(r(s,\alpha )\right)\ll T_{\mu \nu }\left(\widehat{g}\left(r(s,\alpha )\right)\right),$$

(39)

so that the contribution of the quantum CC can be effectively ignored as first approximation. This is physically reasonable, for example, if we treat the case of a stellar-mass Schwarzschild solution. However, the conceptual implication of a stochastic quantum CC on the prescription of the coordinate representation for the background metric tensor, and therefore its intrinsic stochastic character, must be necessarily retained. In the case of a spherically symmetric, static, asymptotically flat geometry, the relevant equation implied by EFE yields the well-known radial-component ODE

$$r{\phi }^{\prime }\left(r\right)+\phi \left(r\right)=1-8\pi r^2\rho (r,{\epsilon }^2,\alpha ),$$

(40)

where

$$r\equiv r(s,\alpha )$$

(41)

represents the stochastic shift (21) and ${\epsilon }^2$ is assumed to be an infinitesimal dimensionless real parameter, $\phi \left(r\right)=-g_{00}\left(r\right)=\frac{1}{g_{rr}\left(r\right)}$ and finally $\rho (r,{\epsilon }^2,\alpha )$ is a suitable (and yet to be prescribed) energy density. Notice that Equation (40) differs from the customary EFE available in the standard approach to GR. The crucial difference is that $r\equiv r(s,\alpha )$ now identifies the stochastic radial-like spherical coordinate. For definiteness, let us now assume that $r(s,\alpha )$ takes the form of an infinitesimal outward stochastic shift performed with respect to the center of symmetry of the BH. This can always be represented in terms of a linear stochastic decomposition of the form

$$\left\{\begin{array}{c}r\equiv r(s,\alpha )=R(s,\alpha )+r_o,\\ r_o=r_S{\epsilon }^2{\alpha }^2,\end{array}\right.$$

(42)

by introducing a stochastic radial shift of the form $r_o=r_S{\epsilon }^2{\alpha }^2$, which depends on the stochastic parameter $\alpha$, where $r_S$ is the Schwarzschild radius (11), while $\epsilon$ is assumed real and infinitesimal and $r\equiv r(s,\alpha )$ denotes the transformed radial coordinate. In addition, $R(s,\alpha )$ is a nonlinear stochastic function such that ${\lim }_{\alpha \to 0}R(s,\alpha )=R$ and which is assumed independent of $r_o$, while R denotes the standard radial spherical coordinate with domain $R\in {\mathbb{R}}^{+}\equiv \left[0,\infty \right]$ and $\alpha \in \left[-1,1\right]$ is an independent and constant stochastic parameter.

Here, a brief comment is in order regarding the assumption of linear shift of the type (42), whose stochastic character has been already discussed following Equation (11) above. This is to dismiss any possible interpretation of CQG-theory in terms of a linear theory of gravity. On the contrary, despite adopting for them the stochastic decomposition (42), it must be stressed that both the background metric tensor and the corresponding quantum modified EFE (36) retain their nonlinear character. In fact, the prescription of the background metric tensor involves ultimately the solution of the fully nonlinear set of quantum-modified Einstein field equations (36). Similarly, for the same reason, the treatment of quantum expectation values of arbitrary quantum observables remains analogously fully nonlinear (see for example Equation (7)).

Next, let us consider the explicit realization of the quantum regularization method. The physically interesting case, in which quantum regularization is expected to occur, is manifestly the one in which $r(s,\alpha )$ remains strictly positive because $r_o>0$, so that the value $r=0$ is ruled out. This occurs if $r_o$ is quadratic, as in the case of Equation (42), in the stochastic parameter $\alpha$ so that the radial displacement $r_o$ remains strictly positive in the whole stochasticity domain (25). This means, in other words, that graviton quantum trajectories cannot reach the center of symmetry of the BH. Consequently, it implies that the energy density $\rho (r,{\epsilon }^2,\alpha )$ is itself by definition stochastic, while $r_o$ represents a stochastic shift of the BH center of symmetry. Notably, however, we stress that the transformation (42) determines a corresponding stochastic displacement both of the center of symmetry and of the BH event horizon. Equation (40) can be formally integrated at once yielding

$$r\phi \left(r\right)-r_o\phi \left(r_o\right)=r-r_o-8\pi {\int }_{r_o}^{r}d{r}^{\prime }{r}^{\prime 2}\rho (r^{\prime },{\epsilon }^2,\alpha ).$$

(43)

Thus, one finds the truncated solution

$$\left\{\begin{array}{c}\phi \left(r\right)=1-\frac{8\pi }{r}{\int }_{r_o}^{r}d{r}^{\prime }{r}^{\prime 2}\rho (r^{\prime },{\epsilon }^2,\alpha ),\\ \phi \left(r_o\right)=1,\end{array}\right.$$

(44)

where $\phi \left(r\right)$ is defined for arbitrary $r\ge r_o$ only. This implies that, provided a BH exists (i.e., the related event horizons are admissible), $\phi \left(r\right)$ is always regular. On the other hand, the question arises whether a BH, in a proper sense, can still exist. To answer this question, let us consider the example provided by the Schwarzschild solution [22]. Similar methods should be used for the Kerr and Reissner–Nordstrom solutions [23,24,25].

5.1. Schwarzschild Case

Let us consider first the case that corresponds to the original treatment by Schwarzschild, namely in which $r_o$ is assumed identically vanishing ($r_o=0$). This corresponds to the special choice for the energy density $\rho (r^{\prime },{\epsilon }^2,\alpha )$ in which its support coincides with the set of the point origin $r=0$, i.e., $\rho (r,{\epsilon }^2,\alpha )$ reduces to the Schwarzschild distribution, namely, a Dirac delta of the form

$$\rho (r,{\epsilon }^2,\alpha )=\rho \left(r\right)\equiv \frac{r_S}{4\pi r^2}\delta \left(r\right).$$

(45)

This delivers at once

$$8\pi {\int }_0^{r}d{r}^{\prime }{r}^{\prime 2}\rho \left(r^{\prime }\right)=r_S,$$

(46)

which yields (on the rhs of Equation (40)) exactly the customary Schwarzschild source term. Incidentally, we notice that the factor $1/r^2$ in front of the Dirac delta in Equation (45) is required in order for the integral (46) to actually exist and be nonvanishing. However, while still keeping the same choice for the energy density, if one assumes, instead, $r_o>0$, one obtains identically

$$8\pi {\int }_{r_o}^{r}d{r}^{\prime }{r}^{\prime 2}\rho (r^{\prime },{\epsilon }^2,\alpha )=0.$$

(47)

Therefore, the effect of the stochastic shift of the type (42) amounts simply to eliminate altogether any $r-$ dependence and thus destroys the BH solution. This means, unavoidably, that in the present case the (actual) physical energy density cannot be assumed of the Schwarzschild type (45). In other words, stochastic effects of the type described by Equation (42) are incompatible with all BH solutions for a choice of point-like source term of the singular type given by Equation (45).

5.2. The Spatially Smeared-Out Gaussian Case

The obvious physical recipe needed to remedy to such inconvenience is to replace the Schwarzschild solution (45) with a spatially smeared-out energy density $\rho (r,{\epsilon }^2,\alpha )$. The latter should be represented by a smooth function which becomes proportional to a Dirac delta in the limit $r_o\to 0,$ i.e., such that

$$\underset{r_{o\to 0}}{lim}\rho (r,{\epsilon }^2,\alpha )=\frac{r_S}{4\pi r^2}\delta \left(r\right).$$

(48)

It is immediate to show that a physically admissible shape for the energy density $\rho (r,{\epsilon }^2,\alpha )$ is provided by a Gaussian distribution. A possible realization, consistent with CQG-theory, is provided by the general prescription

$$\left\{\begin{array}{c}\rho (r,{\epsilon }^2,\alpha )={\rho }_G(r,{\epsilon }^2,\alpha ,\beta )\equiv \frac{r_S^{-2\beta }}{r^{2-2\beta }}{\widehat{\rho }}_G(r,{\epsilon }^2,\alpha ),\\ {\widehat{\rho }}_G(r,{\epsilon }^2,\alpha )=\frac{r_S}{4{\pi }^{3/2}{r}_o}exp\left\{-\frac{r^2}{r_o^2}\right\},\end{array}\right.$$

(49)

where the exponential factor $\beta$ is taken as a constant integer and $r_o$ is defined by Equation (42), which means that Equation (49) is defined only provided $\alpha \ne 0$, a requirement which justifies the choice of the stochastic domain (25). However, the limit (48) requires setting also $\beta =0,$ which recovers exactly the Schwarzschild distribution. This implies the integral conditions

$$8\pi {\int }_{r_o}^{r}d{r}^{\prime }{r}^{\prime 2}{\rho }_G(r^{\prime },{\epsilon }^2,\alpha ,\beta =0)=r_S\left[erf\left(\frac{r}{r_o}\right)-erf\left(1\right)\right],$$

(50)

together with the further conditions

$$\begin{array}{ccc}\hfill 8\pi {\int }_{r_o}^{\infty }d{r}^{\prime }{r}^{{}^{\prime }2}{\rho }_G(r^{\prime },{\epsilon }^2,\alpha ,\beta & =& 0)=\frac{2r_S}{{\pi }^{1/2}}{\int }_1^{\infty }dxexp\left\{-x^2\right\}=r_S\left[1-erf\left(1\right)\right],\hfill \end{array}$$

(51)

$$\begin{array}{ccc}\hfill 8\pi {\int }_{r_o}^{\infty }d{r}^{\prime }{r}^{{}^{\prime }4}{\rho }_G(r^{\prime },{\epsilon }^2,\alpha ,\beta & =& 0)=\frac{2r_S{r}_o^2}{{\pi }^{1/2}}{\int }_1^{\infty }dx{x}^{2}exp\left\{-x^2\right\}=\frac{r_S{r}_o^2}{2}\left[exp(-1)+1-erf\left(1\right)\right].\hfill \end{array}$$

(52)

Incidentally, the same distribution $\widehat{\rho }(r^{\prime },{\epsilon }^2,\alpha )\equiv {\widehat{\rho }}_G(r,{\epsilon }^2,\alpha )$ can be shown to be also the “most likely” energy density distribution, i.e., the one which, in a probabilistic interpretation, satisfies the principle of maximum entropy [26,27], i.e., such that

$$\delta S_{BS}=0$$

(53)

subject to the constraints

$$\left\{\begin{array}{c}8\pi {\int }_{r_o}^{\infty }d{r}^{\prime }\widehat{\rho }(r^{\prime },{\epsilon }^2,\alpha )=r_S\left[1-erf\left(1\right)\right],\\ 8\pi {\int }_{r_o}^{\infty }d{r}^{\prime }{r}^{\prime 2}\widehat{\rho }(r^{\prime },{\epsilon }^2,\alpha )=\frac{r_S{r}_o^2}{2}\left[exp(-1)+1-erf\left(1\right)\right],\end{array}\right.$$

(54)

and where $S_{BS}$ denotes the Boltzmann–Shannon entropy $S_{BS}=-8\pi {\int }_{r_o}^{\infty }d{r}^{\prime }\widehat{\rho }(r^{\prime },{\epsilon }^2,\alpha )ln\widehat{\rho }(r^{\prime },{\epsilon }^2,\alpha )$. On the other hand, a solution of this type is not new and has been earlier proposed with different motivations, for example, in the context of noncommutative geometry [28].

The conclusion is therefore that in the framework of CQG-theory a stochastic shift of the origin of the BH of the type (42) necessarily requires a spatially smeared distribution for the energy density. A possible solution of this type is provided by the Gaussian solution. In particular, a natural choice corresponds to the Gaussian distribution with $\beta =0$ exponential factor. The important remarks to be made are that:

• First, the need for a spatially smeared-out solution for the energy density is a mandatory physical requirement emerging from CQG-theory.
• Then, the Gaussian-type solution (49) exhibits everywhere a regular behavior, simply because the center-origin $r_o$ is effectively unreachable. This occurs because the dimensionless stochastic parameter ${\epsilon }^2/r_S\ll 1$, despite being infinitesimal, is assumed nevertheless strictly positive.
• Furthermore, the truncated solution $\phi \left(r\right)$ given by (44), and to be defined in terms of the same Gaussian-type solution, exhibits everywhere a regular behavior.
• Finally, the classical Schwarzschild singular solution emerges perspicuously from the regular stochastic quantum solution in the limit $r_o\to 0$, realizing explicitly also in the framework of stochastic quantization the peculiar property of CQG-theory of exhibiting an emerging character, namely such that the classical GR solution is consistently implied by the regular quantum one [14].

6. Discussion and Comparison with Literature

In the literature, starting with Bardeen [29], possible models of regularization of singular BH solutions of EFE have been several. Limiting, however, to models based on different QG approaches it is worth mentioning those due to noncommutative geometry [28,30,31,32,33,34], loop quantum gravity [35,36,37,38] and nonlocal quantum gravity [39,40,41].

The comparison with noncommutative geometry models is particularly intriguing for several reasons. The first one is due to the formal analogy of the Gaussian energy density given above (see Equation (49)) with the noncommutative geometry inspired Schwarzschild BH first pointed out in Ref. [31]. The second one arises because at suitably small distances (i.e., sufficiently high energies) graviton particles might/should appear as extended particles. This means that the consequent nonlocal self-interactions (acting on the “different” parts of each graviton particle) are expected to give rise to noncommutative-geometry effects. However, there is a significant departure of CQG-theory (an intrinsically manifestly covariant and therefore local theory), with a noncommutative geometry inspired theory of QG characterized by a minimum-length approach (which, on the contrary, might potentially violate the core concept of covariance). This is actually related to the very notion of minimum length and generalized Heisenberg inequalities arising in such a context [39]. In fact, first, the Heisenberg inequalities (i.e., the generalized uncertainty principle (GUP) [15,16,17,18,19]) typically invoked in minimal length theories do not have an intrinsic 4-tensor character [42]. Second, by definition the minimum length itself is generally not identified with a 4-scalar. As a consequence both notions are in potential contradiction with the requirements set at the basis of CQG-theory, i.e., the principles of covariance and manifest covariance. We should add, furthermore, that the construction of generalized Heisenberg inequalities holding in the context of QG should require: (a) an underlying quantum gravity theory, which satisfies the principle of unitarity, (b) the introduction of a proper notion of scalar product and finally (c) the construction of Heisenberg inequalities expressed in 4-tensor form. Such requirements are generally met only by CQG-theory [11].

Finally—as a brief comment—a comparison with LQG (an intrinsically noncovariant theory cast on a discrete space-time) is possible, although the departure from CQG-theory (a manifestly covariant theory endowed with a continuous space-time) is great. We mention in this connection a series of papers due to Modesto [35,36,37] on the LQG-regularization of singular BH solutions. However, in the context of LQG, regularization is actually a result that emerges perspicuously from elementary considerations. Indeed in the context of LQG the regularization arises as a consequence of the assumed discreteness of the spatial part of space-time [38], i.e., at the expense of losing the continuity of space-time. In contrast, in the context of CQG-theory discussed here, regularization is achieved by taking into account the stochasticity property of quantum graviton trajectories (which characterize the same QG theory) and properly related to the finite-size structure of the energy density (see again Equation (49)).

7. Conclusions

In classical GR the occurrence of space-time singularities, particularly black hole (BH) ones, is a well-known ubiquitous feature [43,44], while the possible regularity of BH solutions is usually simply regarded as a plausible physical requirement. However, in the case of manifestly covariant quantum gravity theory (CQG-theory), the manifestly covariant theory of quantum gravity, such a feature is actually a mandatory prerequisite for the validity of the same theory. The regularity requires that both the covariant and countervariant components of the background metric tensor, when evaluated along arbitrary quantum space-time trajectories, should remain finite. However in quantum gravity, as in quantum mechanics, the parametrization of the theory in terms of quantum trajectories, i.e., in particular both space-time and configuration-space trajectories, is crucial. Such a feature is especially relevant because quantum particles should always be treated as intrinsically nondeterministic, i.e., characterized by the presence of hidden, i.e., stochastic variables.

This means that quantization of classical gravity must properly be adapted to include also the transition from classical (i.e., deterministic) to quantum (i.e., stochastic) space-time trajectories. Such a type of quantization scheme, denoted as stochastic quantization, leads therefore formally to quantum equations that are prescribed, in principle, along arbitrary stochastic quantum trajectories. In particular, starting from the stochastic-modified CQG-quantum wave equation obtained in this way, also an analogous representation has been obtained for the quantum-modified Einstein field equations.

The interesting application considered here concerns the identification of the physical mechanism responsible for the quantum regularization of singular BH solutions, with particular reference to the case of Schwarzschild nonrotating solutions. A number of interesting conclusions emerge:

• The first one, which applies to an in principle arbitrary singular BH solution, is a consequence of the stochastic behavior of quantum space-time trajectories. As a result, the central singularity (at $r^{*}$) of an arbitrary BH solution may actually effectively disappear. This may happen once a “stochastic averaging” operator is acted upon (see Section 4.2).
• The second physical mechanism applies to the specific case of a singular Schwarzschild BH solution. The occurrence of quantum regularization brings about another interesting physical explanation (see Section 5.1). In fact, it is found that the customary Schwarzschild energy density, which is proportional to a Dirac delta localized at $r^{*}$ does not permit to recover the correct physical behavior of the classical Schwarzschild solutions. This conclusion means that a smeared-out stochastic energy density must necessarily be adopted. For this purpose, the case of a stochastic Gaussian-type energy density distribution has been considered. This solution is everywhere regular and exhibit the correct asymptotic behavior of the Schwarzschild solution sufficiently far from the origin point ($r^{*}$). The same distribution has been shown to provide also the “most likely” energy density distribution, i.e., the one which, in a probabilistic interpretation, satisfies a principle of maximum entropy.

Finally, an interesting open question, which is susceptible of possible further developments, emerges following the discussion of Section 3 about the validity of the generalized Heisenberg inequality (see Equations (14) and (19), both holding in the context of CQG-theory). This concerns the possible (rigorous) connection between the concept of minimum length and the phenomenon of quantum regularization of black holes. The topic will be the subject of a forthcoming investigation.

These conclusions are rewarding in particular because they show that the intrinsic stochasticity of quantum trajectories, combined—in the case of the Schwarzschild BH solution—with a smeared-out energy density, may lead to the quantum regularization of the background metric tensor solution for the quantum-modified Einstein field equations.